Well, of course, all of this (Glen and Nick's posts) is ignoring the obvious fact that ambiguity is the antithesis of mathematics. Of course (?!?), there is a nuanced resolution of this tension, having something to do with a difference in worlds between the lofty professor and the practical man, but I'm not sure what it is.
When a teacher asks a student what 2+2 is (hint: 4), the length of the area of a circle with radius 1 (hint: pie), what the integral of a given function is, whether a given number is prime or not, etc. etc. etc., the student doesn't get full credit for saying "Its ambiguous, and the world is better that way!" I doubt anyone would argue that students and lower-level teachers of mathematics are completely wrong in their view that these questions have unambiguous answers. (Though surely some will claim the problems are not adequately specified. For example, is the circle in euclidean space?) So, how do we reconcile claims that ambiguity is at the heart of mathematics with the obvious truth that mathematicians really like producing, teaching, and preaching about unambiguous things? Also, re Glen's post specifically, I think there is value in discriminating between accidental and intentional ambiguity. Not all claims of ambiguity is are claims of ignorance, sometimes situations are actually ambiguous and therefore claims of ambiguity are claims of knowledge. For an example of the former, I may claim that the pitter patter on my roof "May be acorns falling or it may be rain, its ambiguous". In that case, we all agree that it either IS acorns OR rain (while retaining the chance it is both), and it is clear that I am stating my ignorance as to which it is. For an example of the latter, we might ask whether George W.'s "Free Speech Zones" were protecting people's freedom of speech. One possible answer to that question, one that expresses a good understanding of the situation, NOT severe ignorance, might be "In some ways it technically was, but in other ways it severely undermined freedom speech, so the situation is ambiguous." On a lighter note, many jokes an innuendo take advantage of ambiguity, and if you don't think the situation is ambiguous, you won't get it. For example, I once shot an elephant in my pajamas..... what he was doing in my bedroom I'll never know. Eric On Tue, Dec 29, 2009 01:21 PM, "glen e. p. ropella" <[email protected]> wrote: > This perspective is the essential gist of Robert Rosen's message, if you >carve off all the surrounding sophistry. Ambiguity is the essence of >life. If we specialize down into mathematicians, we can say that >ambiguity is the essence of mathematics, as practiced by the animals we >call mathematicians. > >To some extent, this may seem to trivialize what Byers and Bohm are >saying; but I don't think it does. It just places it in a larger context. > >But the paradox Nick points out extends beyond the "mathematics >itself" >question, in tact, up to the "life itself" question. And that brings >me >to my current comment: > >Asserting that ambiguity is the heart of _anything_ is, essentially, >"begging the question" or petitio principii. Ambiguity is just >multi-valued-ness, the ability of a [im]predicate [grin] to take on one >value when evaluated in one context and another value when evaluated in >another context. Hence, ambiguity is (like randomness) a statement of >ignorance. > >So, there are 2 ways to parse the situation (and the quote from Byers) >as a statement of ignorance: > >1) Saying "ambiguity is the heart of math" is saying "we >really don't >understand what we're doing when we do math", or > >2) Saying "ambiguity is the heart of math" is an expression that >math is >a _method_, not knowledge ... an approach, not a thing to be approached. > >Both are compatible with the "mechanism" that Rosen rails about. But >(2) allows us to put off the controversy and continue working together >as holists and reductionists. ... or not. ;-) > > >Quoting Nicholas Thompson circa 09-12-28 10:33 PM: >> Hi, everybody, >> >> The most important part of this message is the first few paragraphs, >don't not read it because it is long. >> >> THE TEXT: >> >> Here are two stimulating quotes from William Byers, How Mathematicians >Think. You will find them on pp 23-25, which happen to be up on Amazon's page >for the book. >> >> Last paragraph of the intro, page 24: >> >> The power of ideas resides in their ambiguity. Thus, any project that >would eliminate ambiguity from mathematics would destroy mathematics. It is >true that mathematicians are motivated to understand, that is, to move toward >clarity, but if they wish to be creative then they must continually go back to >the ambiguous, to the unclear, to the problematic, that is where new >mathematics comes from. Thus, ambiguity, contradiction and their consequences >--conflict, crises, and the problematic-cannot be excised from mathematics. >They are its living heart. >> >> Epigraph from chapter 1, page 25: >> >> "I think people get it upside down when they say the unambiguous is >the reality and the ambiguous merely uncertainty about what is really >unambiguous. Let's turn it around the other way: the ambiguous is the reality >and the unambiguous is merely a special case of it, where we finally manage to >pin down some very special aspect. >> >> David Bohm" >> >> A few pages later, Byers defines ambiguity as involving >> >> "...a single situation or idea that is perceived in two >self-consistent but mutually incompatible frames of reference." >> >> THE SERMON: >> >> Now on the one hand, these passages filled me with joy, because a little >appreciated psychologist of great perspicacity once wrote: >> >> "The insight that science arises from contradiction among concepts is >a useful one for explaining characteristic patterns of birth, growth, and decay >in the sciences. Initially, a phenomenon is brought sharply into focus by its >relationship to a conceptual problem. A first generation of imaginative >investigators is attracted to the phenomenon in the hope of casting light on >the related conceptual issue. These investigators generate a lot of argument, >a little progress, and a lot of publicity. Then a second generation of >scientists attracted, who are drawn to the problem more by the sound of battle >than by any genuine interest in the original issue. By then, the conceptual >issue has been straightened out, the good people have left, and those who >remain devote their time to swirling in ever tighter eddies of technological >perfection. " (Thompson, 1976, My Descent from the Monkey, In P.P.G. >Bateson and P.H. Klopfer (Eds.), Perspectives in Ethology, 2, >221-230. >> >> On the other hand, to call ambiguity the living heart of mathematics seems >a little like calling "mess-making" the living heart of cleaning a >house, or littering the living heart of public sanitation. >> >> It is characteristic of all goals that, if they are achieved, the activity >associated with them ceases. Therefore, for goal directed activity to >continue, it must fail to achieve it's end. But that hardly makes failure the >goal of the activity. >> >> I suspect that Byers may clear this up in subsequent pages, but I thought >it was interesting enough to put it before the group. One way out of the >paradox, lies in Byers's definition's insistence that ambiguity defined by a >contradiction between two clear concepts bound within the same system. If we >understood mathematicians as clarifying the concepts that are bound within a >frame work until their contradiction becomes evident, then the perhaps the >specter of making ambiguity the heart of mathematics becomes less horrifying. > >-- >glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com > > >============================================================ >FRIAM Applied Complexity Group listserv >Meets Fridays 9a-11:30 at cafe at St. John's College >lectures, archives, unsubscribe, maps at http://www.friam.org > > > Eric Charles Professional Student and Assistant Professor of Psychology Penn State University Altoona, PA 16601
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